The homology of simplicial complement and the cohomology of the moment-angle complexes

Abstract

A simplicial complement P is a sequence of subsets of [m] and the simplicial complement P corresponds to a unique simplicial complex K with vertices in [m]. In this paper, we defined the homology of a simplicial complement $H_{i,\sigma}(\Lambda^{*,*}[P], d)$ over a principle ideal domain k and proved that $H_{*,*}(\Lambda[P], d)$ is isomorphic to the Tor of the corresponding face ring k(K) by the Taylor resolutions. As applications, we give methods to compute the ring structure of Tor_{*,*}^{k[x]}(k(K), k)$, $link_{K}\sigma$, $star_{K}\sigma$ and the cohomology of the generalized moment-angle complexes.